How to snap plane center to a static vertex?
It looks as if you've already done most of this, but for the sake of completeness, trying to keep the steps minimal from the worst case:

To place the tetrahedron's origin at its centroid


In Object Mode, tetrahedron selected, ShiftAltO > 'Origin to Center of Mass'


To place the cone's origin at its point


In Edit mode, select the point vertex
ShiftS > 'Cursor to Selected'
In Object Mode, ShiftAltO > 'Origin to 3D Cursor'


To place the cone's point at the centroid of the terahedron


In Object Mode, tetrahedron selected, ShiftS > 'Cursor to Selected'
Cone selected, ShiftS > 'Selection to Cursor'


TL;DR, cut to the chase....

To align the cone to one of the vertices of the tetrahedron

(given you have created a vertex at the center of the cone's base, and having centered the cone to the tetrahedron, as above)


Set Pivot to '3D Cursor', Snap to 'Active' and 'Vertex'.
Object Mode, ShiftS 'Cursor to Selected' ( either cone or tetrahedron, to put the cursor at the cone's point) 
Edit Mode, select all vertices of the cone, make the base-center vertex active. 
In each axis in turn, RX,RY,RZ, snap the active vertex in rotation to the desired corner of the tetrahedron.


To move the base of the cone to one of the vertices of the tetrahedron


With preceding step taken, S scale the base to the corner, snapping to the active base-center vertex.


(You don't actually have to create a vertex at the center of the base of the cone .. instead of setting the snap to 'Active' you can set it to 'Center' and snap the cone's base as a whole, to the corner of the tetrahedron.)